Question

Scientists shine a laser beam on a $35-\mu \mathrm{m}$-wide slit and produce a diffraction pattern on a screen $70\mathrm{cm}$behind the slit. Careful measurements show that the intensity first falls to $25\mathrm{\%}$ of maximum at a distance of $7.2\mathrm{mm}$ from the center of the diffraction pattern. What is the wavelength of the laser light?

Short answer

The wavelength of the laser light$\lambda =596\mathrm{nm}$

Step-by-step solution

Intensity

The parameters derived the luminosity as a function of y.

$I_{\text{slit}}=I_0{\left[\frac{\sin (\pi ay/\lambda L)}{\pi ay/\lambda L}\right]}^2$

$\frac{I}{I_o}=0.25={\left[\frac{\sin (\pi ay/\lambda L)}{\pi ay/\lambda L}\right]}^2$

$\frac{\sin (\pi ay/\lambda L)}{\pi ay/\lambda L}=\sqrt{0.25}=0.5$ (Rearrange)

Graphing method

Recall that we desire to determine, but that the sin and integer both have the same term, we have a mystical equation that must be solved by trial and error or by graph. To simplify the equation, we implement the following substitution.

$\begin{array}{r}\frac{\sin \left(x\right)}{x}=0.5\\ \sin \left(x\right)=0.5x\end{array}$

Now you'll be able to see by the equation above, we're trying to find the purpose where the worth equals the worth, or the spot where the graphs of two functions meet. All we've got to try and do now could be draw and find out the midpoint, which I wiped out Desmos, as shown by the image below.

Transcendental Equation

So we obtained x=1.895 as the precise point x which fits the transcendental equation. Hence

$\frac{\pi ay}{\lambda L}=1.895$

$$\begin{gathered} \lambda =\frac{\pi ay}{1.895L} \\ =\frac{\pi \left(35\times {10}^{-6}\mathrm{m}\right)\times 7.2\times {10}^{-3}\mathrm{m}}{1.895\times 0.7\mathrm{m}} \end{gathered}$$

$\lambda =596\mathrm{nm}$